Advertisement

Abstract

Nullnorms have been produced from triangular norms and triangular conorms and they have several applications in fuzzy logic. The main purpose of this paper is to study the order induced by nullnorms on bounded lattices. We discuss the relationship between the natural order and the order induced by a nullnorm on bounded lattice.

References

  1. 1.
    Aşıcı, E.: An order induced by nullnorms and its properties. Fuzzy Sets Syst. (in press). doi: 10.1016/j.fss.2016.12.004
  2. 2.
    Aşıcı, E., Karaçal, F.: On the \(T\)-partial order and properties. Inf. Sci. 267, 323–333 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aşıcı, E., Karaçal, F.: Incomparability with respect to the triangular order. Kybernetika 52, 15–27 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)Google Scholar
  5. 5.
    Calvo, T., De Baets, B., Fodor, J.: The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. 120, 385–394 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Casasnovas, J., Mayor, G.: Discrete t-norms and operations on extended multisets. Fuzzy Sets Syst. 159, 1165–1177 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Baets, B., Mesiar, R.: Triangular norms on the real unit square. In: Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain, pp. 351–354 (1999)Google Scholar
  8. 8.
    Çaylı, G.D., Karaçal, F., Mesiar, R.: On a new class of uninorms on bounded lattices. Inf. Sci. 367–368, 221–231 (2016)CrossRefGoogle Scholar
  9. 9.
    Çaylı, G.D., KaraÇal, F.: Construction of uninorms on bounded lattices. Kybernetika 53, 394–417 (2017)Google Scholar
  10. 10.
    Çaylı, G.D.: On a new class of t-norms and t-conorms on bounded lattices. Fuzzy Sets Syst. (in press) doi: 10.1016/j.fss.2017.07.015
  11. 11.
    Drewniak, J., Drygaś, P., Rak, E.: Distributivity between uninorms and nullnorms. Fuzzy Sets Syst. 159, 1646–1657 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Drygaś, P.: A characterization of idempotent nullnorms. Fuzzy Sets Syst. 145, 455–461 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karaçal, F., Ince, M.A., Mesiar, R.: Nullnorms on bounded lattice. Inf. Sci. 325, 227–236 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Karaçal, F., Kesicioğlu, M.N.: A T-partial order obtained from t-norms. Kybernetika 47, 300–314 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  16. 16.
    Liang, X., Pedrycz, W.: Logic-based fuzzy networks: a study in system modeling with triangular norms and uninorms. Fuzzy Sets Syst. 160, 3475–3502 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mas, M., Mayor, G., Torrens, J.: t-operators. Int. J. Uncertain. Fuzz Knowl.-Based Syst. 7, 31–50 (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms and t-operators. Fuzzy Sets Syst. 128, 209–225 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. 8, 535–537 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mitsch, H.: A natural partial order for semigroups. Proc. Am. Math. Soc. 97, 384–388 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saminger, S.: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets Syst. 157, 1403–1416 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xie, A., Liu, H.: On the distributivity of uninorms over nullnorms. Fuzzy Sets Syst. 211, 62–72 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Software Engineering, Faculty of TechnologyKaradeniz Technical UniversityTrabzonTurkey

Personalised recommendations